OSR My details: -> Aditya kiran ->Grad-1 st year Applied Math ->UnderGrad- Major in Information technology

HILBERT’S 23 PROBLEMS

Hilbert’s 23 problems David Hilbert was a German mathematician. He published 23 problems in They were all unsolved at that time and were quite important for 20 th century Mathematics. He was also a physicist.. Hilbert spaces named after him Co-discoverer of general relativity..

SOLVED PARTIALLY RESOLVED UNSOLVED

All these questions and topics are highly researched since the last 100 years. So it might be difficult to understand some of them without pre-knowledge. So, I wil try to convey whatever I’d understood.

1. Cantor's continuum hypothesis “There is no set whose cardinality is strictly between that of the integers and that of the real numbers” Cardinality is the number of elements of a set. But when it comes to finding the size of infinite sets, the cardinality can be a non-integer. The hypothesis says that the cardinality of the set of integers is strictly smaller than that of the set of real numbers So there is no set whose cardinality is between these two sets. PARTIALLY SOLVED

2. Consistency of arithmetic axioms In any proof in arithmetic, Can we prove that all the assumptions and statements are consistent? Is arithmetic free of internal contradiction.? PARTIALLY SOLVED

3. Polyhedral assembly from polyhedron of equal volume Given 2 polyhedra of same volume. Now the 1 st one is broken up into finitely many parts. Now Can we join those broken parts to form the 2 nd polyhedron.?? i.e Can we decompose 2 polyhedron identically? NO!! SOLVED

4. Constructibility of metrics by geodesics Construct all metrics where the lines are geodesics. Geodesics are straightlines on curves spaces. Find geometries on geodesics whose axioms are close to euclidean geometry (with the parallel postulate removed.etc) Solved by G. Hamel. PARTIALLY SOLVED

5. Are continuous groups automatically differential groups? Existence of topological groups as manifolds that are not differential groups. Is it always necessary to assume differentiability of functions while defining continuous groups? NO.!.. PARTIALLY SOLVED A Lie group

6. Axiomatization of physics Mathematical treatment of the axioms of physics. Says that all physical axioms and theories need a strong mathematical framework. It is desirable that the discussion of the foundations of mechanics be taken up by mathematicians also. Eg: A point is an object without extension. Laws of conservation (Δε(a,b) = ΔK(a,b) + ΔV(a,b) = 0) The total inertial mass of the universe is conserved…etc Time is quantized NOT SOLVED

7. Genfold-Schneider theorem Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? YES.!!  Transcendental number=> -not algebraic -not a root of polynomial with rational coeffs. Eg: ∏,e..etc SOLVED

8. Riemann hypothesis Reg the location of non-trivial roots of the Riemann-zeta function. Riemann said that, ”the real-part of the non- trivial roots is always =1/2” This has implications on: -Prime number distribution -Goldbach conjecture NOT SOLVED

On Prime numbers: Riemann proposed that the magnitudes of oscillation of primes around their expected position is controlled by the real-part of the roots of the zeta function. Prime number thrm=> :- ∏(x)

GoldBach conjecture: Every even integer greater than 2 can be expressed as sum of two primes

9. Algebraic number field reciprocity theorem Find the most general law of reciprocity thrm in any algebric number fields. Eg: quadratic reciprocity: p,q are distinct odd no.s PARTIALLY SOLVED

10. Matiyasevich's theorem Solved Does there exist some algorithm to say if a polynomial with integer co-effs has integer roots? Does there exist an algorithm to check if a diophantine equation can have integer co-effs. -Diophantine eqn is a polynomial that takes only integer values for variables SOLVED

11. Quadratic form solution with algebraic numerical coefficients Solving quadratic forms with Algebric numeric co-efficients. Improve theory of quadratic forms like ax 2 +bxy+cy 2.,etc PARTIALLY SOLVED

12. Extension of Kronecker's theorem to other number fields Extend Kronecker's problem on abelian extensions of rational numbers. Statement: “ every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field “ NOT SOLVED

13. Solution of 7th degree equations with 2-parameter functions Take a general 7 th degree equation x 7 +ax 3 +bx 2 +cx+1=0. Can its solution as a function of a,b,c be expressed using finite number of 2-variable functions Can every continuous function of three variables be expressed as a composition of finitely many continuous functions of two variables PARTIALLY SOLVED

14. Proof of finiteness of complete systems of functions Are rings finitely generated? Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? SOLVED

15. Schubert's enumerative calculus Require a rigorous foundation of Shubert’s enumerative calculus. enumerative calculus=> counting problem of projective geometries PARTIALLY SOLVED

16. Problem of the topology of algebraic curves and surfaces Describe relative positions of ovals originating from a real algebraic curves as a limit-cycles of polynomial vector field. Limit cycle NOT SOLVED

17. Problem related to quadratic forms Given a multivariate polynomial that takes only non-negative values over the reals, can it be represented as a sum of squares of rational functions? A rational function is any function which can be written as the ratio of two polynomial functions Eg: SOLVED

18. Existence of space-filling polyhedron and densest sphere packing The 18 th question asks 3 questions: a)Symmetry groups in n-dimensions Are there infinitely many essential sub-groups in n-D space? b)Anisohedral tiling in 3 dimensions Does there exist an anisohedral polyhedron in 3D euclidean space? c)Sphere packing SOLVED

19. Existence of Lagrangian solution that is not analytic Are the solutions of lagrangians always analytic.? – YES SOLVED

20. Solvability of variational problems with boundary conditions Do all boundary value problems have solutions.? SOLVED

21. Existence of linear differential equations with monodromic group Proof of the existence of linear differential equations having a prescribed monodromic group monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity SOLVED

22. Uniformization of analytic relations It entails the uniformization of analytic relations by means of automorphic functions. SOLVED

23. Calculus of variations Develop calculus of variations further. The 23 rd question is more of an encouragement to develop the theory further. NOT SOLVED

So these were the 23 problems that Hilbert had proposed for the 20 th century mathematicians..

Apart from these there are another class of problems called the ‘’Millenium problems’’ A set of 7-problems Published in 2000 by Clay Mathematics Institute. Only 1 out of 7 are solved till date. 1 7

The seven Millenium problems are:  P versus NP problem  Hodge conjecture  Poincaré conjecture ----(solved)  Riemann hypothesis  Yang–Mills existence and mass gap  Navier–Stokes existence and smoothness  Birch and Swinnerton-Dyer conjecture

Poincaré conjecture Statement: “ Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.”  Grigori Perelman, a Russian mathematician it solved in 2003 Grigori Perelman  He was selected for the Field prize and the Millenium prize.  He declined both of them, saying that he is not interested In money or fame

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